Semisimple ring

Remarks

Example: for a field $F$, the ring $R = M_2(F)$ is semisimple. The left ideals $L_1$ and $L_2$ of matrices that vanish outside the first and the second column are minimal, and $R = L_1 \oplus L_2$.

Non-example: $\Z$ is not semisimple. A minimal nonzero ideal of $\Z$ would be some $n\Z$ with $n \ne 0$, but $2n\Z$ lies strictly inside it, so $\Z$ has no minimal nonzero ideal.

A ring is left semisimple if and only if it is right semisimple, so one simply says semisimple. Equivalent conditions: every left $R$-module is semisimple; every left $R$-module is projective; $R$ is left Artinian and its Jacobson radical is $0$. A semisimple ring is a finite direct sum of minimal left ideals, because $1$ lies in a finite partial sum.